Question: Solve for $n$. $\left(x^3\right)^{3}=x^n$ $n=$
The general rule for powers of powers is $\left(x^m\right)^{n}=x^{m\cdot n}$. Let's expand the powers for $ \left({x^3}\right)^{{3}}=x^n}$. $\begin{aligned} \left({x^3}\right)^{3}&=\underbrace{{x^3\cdot x^3 \cdot x^3}}_{\text{3 times}} \\\\\\ &=\underbrace{ \underbrace{{x\cdot x \cdot x}}_\text{3 times} \cdot \underbrace{{x\cdot x \cdot x}}_\text{3 times} \cdot \underbrace{{x\cdot x \cdot x}}_\text{3 times}} _{\text{3 times}} \\\\ &=\underbrace{x\cdot x\cdot x\cdot x\cdot x\cdot x\cdot x\cdot x\cdot x}_{n\text{ times}}} \\\\ \end{aligned}$ $n = 9$